International Journal of

Transcription

1 Iter. J. Fuzzy Mathematcal Archve Vol. 3, 203, 36-4 ISSN: (P), (ole) Publshed o 7 December Iteratoal Joural of Mult Objectve Fuzzy Ivetory Model Wth Demad Depedet Ut Cost, Storage Space ad Lead Tme Costrats A Karush Kuh Tucker Codtos Approach S.Ragaayak ad C.V.Seshaah Departmet of Mathematcs Sr Ramakrsha Egeerg College Combatore, Tamladu, Ida Emal: rakhul07@yahoo.com; cvseshaah@gmal.com Receved 24 November 203; accepted 6 December 203 Abstract. A mult-objectve vetory model wth demad depedet ut cost ad leadg tme has bee formulated wth storage space, umber of orders ad producto cost as costrats. I most of the real world stuatos the cost parameters the objectve fucto ad costrats of the decso makers are mprecse ature. A demad depedet ut cost s assumed ad solved usg Karush Kuh Tucker codtos. Here the ut producto cost s cosdered uder fuzzy evromet. The model has bee solved wth demad, lot sze ad leadg tme as decso varables. The model s llustrated for a sgle tem. Keywords: Ivetory, Membershp Fucto, Karush -Kuh-Tucker Codto, demad depedet, lead tme AMS Mathematcs Subject Classfcato (200): 90B05. Itroducto I most of the exstg lterature, vetory related costs are assume to be determstc ad represeted as real umbers. But, real stuato the vetory costs are usually mprecse ature due to the fluece of varous ucotrollable factors. For example, costs may deped o some foreg moetary ut. I such a case, due to exchage rates, the costs are ofte ot kow precsely. Ivetory carryg cost may also depedet o some parameters lke terest rate ad stock keepg ut s market prce, whch are ot precse. Also the shortage cost s ofte dffcult to determe precsely the case whe t reflects ot just lost sale but also a loss of customers wll. Therefore, these cost parameters are descrbed as approxmately equal some certa amout ad so t s more reasoable to characterze these parameters as fuzzy. Sce the developmet of EOQ model by Harrs [], lot of research works have bee carred out vetory cotrol system. I the exstg lterature, vetory models are 36

2 Mult Objectve Fuzzy Ivetory Model Wth Demad Depedet Ut Cost, Storage Space ad Lead Tme Costrats A Karush Kuh Tucker Codtos Approach geerally developed uder the assumpto of costat or stochastc lead-tme. A umber of research papers have already bee publshed ths drecto Das.C [2], ad Foote et.al [3] etc). Recetly, Kalpakam ad Sapa [4] studed a pershable vetory model wth stochastc lead-tme. But real lfe stuatos, the lead-tme s ormally vague ad mprecse.e. ucerta o-stochastc sese. It wll be more realstc to cosder the lead tme as fuzzy ature. I geeral the classcal vetory problems are desged by cosderg that the demad rate of a tem s costat ad determstc ad that the ut prce of a tem s cosdered to be costat ad depedet ature. But practcal stuato, ut prce ad demad rate of a tems may be related to each other. Whe the demad of a tem s hgh, a tem s produced large umbers ad fxed cost of producto are spread over a large umber of tems. Hece the ut cost of the tem decreases..e., the ut prce of a tem versely relates to the demad of that tem. So demad rate of a tem may be cosdered as a decso varable. I mult-objectve mathematcal programmg problems, a decso maker s requred to maxmze/mmze two or more objectves smultaeously over a gve set of possble stuatos. I the crsp evromet, all parameters the total vetory cost such as holdg cost, orderg cost, set-up cost, purchasg cost, deterorato rate, demad rate ad producto rate etc. are kow ad have defte value wthout ambguty [5]. Some of the busess stuatos ft such codtos, but most of the stuatos ad the day-byday chagg market scearo the parameters ad varables are hghly ucerta or mprecse. Ths paper develops a vetory problem wth demad depedet ut rate for a prescrbed fte tme horzo allowg mprecse lead-tme. A demad depedet ut cost had bee treated by some researchers the problem of EOQ model. Chag [5] studed a EOQ model wth demad depedet ut cost of sgle tem. Be-Daya ad Abdul Raouf [6] descrbed the problem of vetory models volvg lead tme as a decso varable. Abou-et-Ata ad Kotb[7] developed a crsp vetory model uder two restrctos. Also Teg ad Yag [8] examed determstc vetory lot sze model wth tme varyg demad ad cost uder geeralzed holdg costs. Other related studes were wrtte by Jag & Kle [9].The cocept of fuzzy set theory was frst troduced by Zadeh [0]. Later o Bellma ad Zadeh [] used the fuzzy set theory to the decso makg problems. Hece Fuzzy set theory has made o etry to the vetory cotrol system. May researchers solved fuzzy mult tem mult objectve vetory especally usg geometrc programmg method [2,3,4]. Here we solve the model usg Karush-Kuh-Tucker Codtos wth ut producto cost uder fuzzy evromet. 2. Notatos ad assumptos To costruct the model, we defe the followg otatos D = Aual demad rate (decso varable) p = ut purchase(producto) cost S = orderg cost 37

3 S.Ragaayak ad C.V.Seshaah H = ut holdg( Ivetory carryg)cost per ut tem Ss= Kσ L = Safety stock = umber of dfferet tems (carred vetory) L = Leadg rate tme (decso varable) Q = Producto (order) quatty take (decso varable) TC( D, Q, L ) = Average aual total cost for the th tem w = storage space per tem W = Floor or Shelf Space avalable B = Total vestmet cost for repleshmet t = umber of orders Assumptos The followg basc assumptos are made the model () Tme horzo s fte (2) No shortages are allowed (3) Ut producto cost s versely related to the demad rate. (.e) p = A D β =,2,3, A > 0, β Where A,β are real costats,selected to provde the best ft of the estmated cost fucto. (4) Lead tme crashg cost s related to the lead tme by a fucto of the form b R(L )= α L, =,2,..., α >0, 0 < b 0.5 (5) Objectve s to mmze the aual relevat total cost. 3. Mathematcal formulato The aual relevat total cost [sum of producto, order, vetory carryg ad lead tme crashg costs]whch accordg to the basc assumptos of the EOQ model s: SD Q D TC( D, Q, L ) = pd + + Kσ L H R( L ) = Q Q () Substtutg p ad R(L ) () gves β S D Q D b TC( D, Q, L ) = AD + + Kσ L H αl = Q Q To derve the optmal total cost a vetory problems,there are some restrctos o avalable resourses. () There s a lmtato o the avalable warehouse floor space where the tems are () to be stored..e. = w Q W Ivestmet amout o total producto cost caot be fte, t may have a upper lmt o the maxmum vestmet.e. = p Q B 38

4 Mult Objectve Fuzzy Ivetory Model Wth Demad Depedet Ut Cost, Storage Space ad Lead Tme Costrats A Karush Kuh Tucker Codtos Approach () or = β AD Q B A upper lmt o the umber of orders that ca be made a tme cycle D o the system,.e. t. Q = 4. Fuzzy vetory model The above crsp model uder fuzzy evromet wth p s as fuzzy decso varable, reduces to β D Q D b MTC( D, Q, L, p ) = AD + S + Kσ L H αl = Q Q Subject to the costrats = = w Q W β AD Q B D t = Q [Here cap deotes the fuzzfcato of the parameter] 5. Membershp fucto A membershp fucto for the fuzzy varable p s defed as follows, p L L U L p µ p ( x) =, L L p U L U L L L 0, p U L Here U ad L are upper lmt ad lower lmt of p respectvely. L L 6. Numercal example The decso varables amely the optmal order quatty Q, optmal demad rate D ad optmal lead tme L whose values determe the mmum aual relevat total cost are computed for dfferet values of β. The parameters of the model are show Table - Assume the stadard devato σ =6 ut/year, K=2, 3 p 8 The optmal values of the producto batch Q,demad rate D, lead tme L ad mmum total cost are gve Table 2. 39

5 S.Ragaayak ad C.V.Seshaah S A H α w W B T 200$ 5 0.8$ Table : The optmal soluto of Q, D ad L as a fucto of β a β b D Q L M TC p µ p x x x x x Table 2: The optmal soluto s D =.62, Q = 28.58, L =.3x0-5 ad M TC = $ whch correspods to maxmum membershp fucto It has bee see that as β value creases, the lot sze Q, the demad D, lead tme L creases whereas the mmum total cost decreases. 7. Cocluso I ths paper we have proposed a cocept of the optmal soluto of the vetory problem wth fuzzy cost prce per ut tem. A vetory model wth demad depedet ut cost ad lead tme depedet of o leadg tme crashg cost wth lmted lot sze, warehouse ad vestmet s solved usg Karush-Kuh-Tucker Codtos. Here the optmal soluto s calculated wth fuzzy ut prce per tem. The result revels the mmum expected aual total cost of the vetory model. The model ca be exteded for more tha oe tem. Also t ca be solved for varous costrats lke lmted budgetary, set up cost etc. REFERENCES. F.Harrs, Operatos ad cost, Factory Maagemet Servce, Chcago, A.W.Shaw Co., C.Das, Effect of lead tme o vetory; a statc aalyss, Operato Research, 26 (975)

6 Mult Objectve Fuzzy Ivetory Model Wth Demad Depedet Ut Cost, Storage Space ad Lead Tme Costrats A Karush Kuh Tucker Codtos Approach 3. B.Foote, N.Kebrac ad H.Kum, Hurrstc polces for vetory orderg problems wth log ad radom varyg lead tmes, Joural of Operatos ad Maagemet, 7 (988) S.Kalpakam ad K.P.Sapa, Pershable system wth stochastc lead tme, Math. Computer Modelg, 2 (995) T.C.E. Cheg, A ecoomc order quatty wth demad-depedet ut cost, Europea Joural of Operatoal Research, 40(2) (989) M. Be-Daya ad A. Raouf, Ivetory models volvg lead tme as a decso varable, Joural of the Operatoal Research Socety, 45(5) (994) M.O.Abou-El-Ata ad K.A.M.Kotb, Mult-tem EOQ vetory model wth varyg holdg cost uder two restrctos: a geometrc programmg approach, Producto Plag ad Cotrol, 8(6) (997) J.T.Teg ad H.L.Yag, Determstc vetory lot-sze models wth tme-varyg demad ad cost uder geeralzed holdg costs, Iformato ad Maagemet Sceces, 8(2) (2007) H.Jug ad C.M.Kle, Optmal vetory polces uder decreasg cost fuctos va geometrc programmg, Europea Joural of Operatoal Research, 32(3) (200) L.A.Zadeh, Fuzzy sets, Iform. ad Cotrol, 8 (965) R.E.Bellma ad L.A.Zadeh, Decso makg a fuzzy evromet, Maagemet Scece, 7(4) (970) B4-B N. K. Madal, T. K. Roy ad M. Mat, Ivetory model of deterorated tems wth a costrats: a geometrc programmg approach, Europea Joural of Operatoal Research, 73() (2006) K.A.M.Kotb ad H. A. Fergay, Mult-tem EOQ model wth varyg holdg cost :a geometrc programmg approach, Iteratoal Mathematcal Forum, 6(23) (20) K.A.M.Kotb ad Hala A.Fergacy, Mult tem EOQ model wth both demaddepedet ut cost ad varyg lead tme va geomatc programmg, Appled Mathematcs, 2 (20) P.K.Gupta, Ma Moha, Problems Operatos Research (Methods ad Solutos), S. Chad Co., (2003)